Neighbourhood (mathematics)

A set in the plane is a neighbourhood of a point if a small disk around is contained in .

A rectangle is not a neighbourhood of any of its corners.

In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. Intuitively speaking, a neighbourhood of a point is a set containing the point where one can move that point some amount without leaving the set.

If is a subset of then a neighbourhood of is a set that includes an open set containing . It follows that a set is a neighbourhood of if and only if it is a neighbourhood of all the points in . Furthermore, it follows that is a neighbourhood of iff is a subset of the interior of .

The above definition is useful if the notion of open set is already defined. There is an alternative way to define a topology, by first defining the neighbourhood system, and then open sets as those sets containing a neighbourhood of each of their points.

A neighbourhood system on is the assignment of a filter (on the set ) to each in , such that

the point is an element of each in

each in contains some in such that for each in , is in .

One can show that both definitions are compatible, i.e. the topology obtained from the neighbourhood system defined using open sets is the original one, and vice versa when starting out from a neighbourhood system.

A Deleted neighbourhood of a point (sometimes called a punctured neighbourhood) is a neighbourhood of , without . For instance, the interval is a neighbourhood of in the real line, so the set is a deleted neighbourhood of . Note that a deleted neighbourhood of a given point is not in fact a neighbourhood of the point. The concept of deleted neighbourhood occurs in the definition of the limit of a function.